Dirichlet series
| L(s) = 1 | − 2-s − 4-s + 3·8-s + 9-s − 4·11-s − 4·13-s − 16-s + 8·17-s − 18-s + 8·19-s + 4·22-s + 8·23-s + 25-s + 4·26-s + 4·31-s − 5·32-s − 8·34-s − 36-s − 20·37-s − 8·38-s + 12·41-s + 8·43-s + 4·44-s − 8·46-s + 16·47-s − 14·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 1/4·16-s + 1.94·17-s − 0.235·18-s + 1.83·19-s + 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.784·26-s + 0.718·31-s − 0.883·32-s − 1.37·34-s − 1/6·36-s − 3.28·37-s − 1.29·38-s + 1.87·41-s + 1.21·43-s + 0.603·44-s − 1.17·46-s + 2.33·47-s − 2·49-s − 0.141·50-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(14400\) = \(2^{6} \cdot 3^{2} \cdot 5^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.918156\) |
| Root analytic conductor: | \(0.978879\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 14400,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6643762671\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6643762671\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|---|
| bad | 2 | $C_2$ | \( 1 + T + p T^{2} \) | |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.7.a_o |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.e_w | |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.13.e_be | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.17.ai_bu | |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | 2.19.ai_cc | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.23.ai_bu | |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.29.a_cc | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.31.ae_ck | |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) | 2.37.u_gs | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.41.am_dy | |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | 2.43.ai_dy | |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) | 2.47.aq_gc | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.53.i_di | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.59.m_fu | |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.61.a_eo | |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) | 2.67.ay_ks | |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.a_da | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.73.e_g | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.79.m_gc | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.83.aq_ig | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.89.u_kc | |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.97.ae_hq | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.9590260302, −15.8447679316, −15.4405745228, −14.4242447018, −14.1407044070, −14.0069258505, −13.0980888029, −12.6461787614, −12.3465554571, −11.7481867694, −10.7356813406, −10.6789224512, −9.93911187018, −9.52345167581, −9.19190896009, −8.32936117372, −7.66488013442, −7.50925237905, −6.93140021141, −5.55946263088, −5.23920392625, −4.73356037049, −3.51487045961, −2.78204474347, −1.14288269174, 1.14288269174, 2.78204474347, 3.51487045961, 4.73356037049, 5.23920392625, 5.55946263088, 6.93140021141, 7.50925237905, 7.66488013442, 8.32936117372, 9.19190896009, 9.52345167581, 9.93911187018, 10.6789224512, 10.7356813406, 11.7481867694, 12.3465554571, 12.6461787614, 13.0980888029, 14.0069258505, 14.1407044070, 14.4242447018, 15.4405745228, 15.8447679316, 15.9590260302